Found problems: 1581
(A.Myakishev) Let triangle $ A_1B_1C_1$ be symmetric to $ ABC$ wrt the incenter of its medial triangle. Prove that the orthocenter of $ A_1B_1C_1$ coincides with the circumcenter of the triangle formed by the excenters of $ ABC$.
In a triangle $ABC$, the inner and outer bisectors of the $\angle A$ meet the line $BC$ at $D$ and $E$, respectively. Let $d$ be a common tangent of the circumcircle $(O)$ of $\triangle ABC$ and the circle with diameter $DE$ and center $F$. The projections of the tangency points onto $FO$ are denoted by $P$ and $Q$, and the length of their common chord is denoted by $m$. Prove that $PQ = m$
In triangle $ ABC$, $ AC \equal{} 13, BC \equal{} 14,$ and $ AB\equal{}15$. Points $ M$ and $ D$ lie on $ AC$ with $ AM\equal{}MC$ and $ \angle ABD \equal{} \angle DBC$. Points $ N$ and $ E$ lie on $ AB$ with $ AN\equal{}NB$ and $ \angle ACE \equal{} \angle ECB$. Let $ P$ be the point, other than $ A$, of intersection of the circumcircles of $ \triangle AMN$ and $ \triangle ADE$. Ray $ AP$ meets $ BC$ at $ Q$. The ratio $ \frac{BQ}{CQ}$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\minus{}n$.
Show that $ \rho (f)$ changes continously over $ f$. It means for every bijection $ f: S^1\rightarrow S^1$ and $ \epsilon > 0$ there is $ \delta > 0$ such that if $ g: S^1\rightarrow S^1$ is a bijection such that $ \parallel{}f \minus{} g\parallel{} < \delta$ then $ |\rho(f) \minus{} \rho(g)| < \epsilon$.
Note that $ \rho(f)$ is the rotatation number of $ f$ and $ \parallel{}f \minus{} g\parallel{} \equal{} \sup\{|f(x) \minus{} g(x)| | x\in S^1\}$.
(i) 15 chairs are equally placed around a circular table on which are name cards for 15 guests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated.
(ii) Give an example of an arrangement in which just one of the 15 guests is correctly seated and for which no rotation correctly places more than one person.
Let $P$ be an arbitrary variable point in the plane of a triangle $ABC. A_1$ is the projection of $P$ onto $BC, A_2$ is the midpoint of line segment $PA_1, A_2P$ meets $BC$ at $A_3, A_4$ is the reflection of $P$ about $A_3$. Prove that $PA_4$ has a fixed point.
Trần Quang Hùng
Line $PQ$ is tangent to the incircle of triangle $ABC$ in such a way that the points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively. On the sides $AB$ and $AC$ are selected points $M$ and $N$, respectively, so that $AM = BP$ and $AN = CQ$. Prove that all lines constructed in this manner $MN$ pass through one point
In an acute-angled triangle $ABC$ the angle at $B$ is greater than $45^\circ$. Points $D,E, F$ are the feet of the altitudes from $A,B,C$ respectively, and $K$ is the point on segment $AF$ such that $\angle DKF = \angle KEF$.
(a) Show that such a point $K$ always exists.
(b) Prove that $KD^2 = FD^2 + AF \cdot BF$.
In a triangle ABC with circumcentre O and centroid M, lines OM and AM are
perpendicular. Let AM intersect the circumcircle of ABC again at A′. Let lines BA′ and AC intersect at D and let lines CA′ and AB intersect at E. Prove that the circumcentre of triangle ADE lies on the circumcircle of ABC.
Let $ ABC$ be a non-isosceles triangle, in which $ X,Y,$ and $ Z$ are the tangency points of the incircle of center $ I$ with sides $ BC,CA$ and $ AB$ respectively. Denoting by $ O$ the circumcircle of $ \triangle{ABC}$, line $ OI$ meets $ BC$ at a point $ D.$ The perpendicular dropped from $ X$ to $ YZ$ intersects $ AD$ at $ E$. Prove that $ YZ$ is the perpendicular bisector of $ [EX]$.
Let $ABC$ and $PQR$ be two triangles such that
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[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
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Prove that $AB+AC=PQ+PR$.
Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively,
such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively.
Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$.
Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$
Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$.
[i]Proposed by David Stoner[/i]
Points $D,E,F$ are on the sides $BC, CA$ and $AB$, respectively which satisfy $EF || BC$, $D_1$ is a point on $BC,$ Make $D_1E_1 || D_E, D_1F_1 || DF$ which intersect $AC$ and $AB$ at $E_1$ and $F_1$, respectively. Make $\bigtriangleup PBC \sim \bigtriangleup DEF$ such that $P$ and $A$ are on the same side of $BC.$ Prove that $E, E_1F_1, PD_1$ are concurrent.
[color=red][Edit by Darij: See my post #4 below for a [b]possible correction[/b] of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?][/color]
let the incircle of a triangle ABC touch BC,AC,AB at A1,B1,C1 respectively. M and N are the midpoints of AB1 and AC1 respectively. MN meets A1C1 at T . draw two tangents TP and TQ through T to incircle. PQ meets MN at L and B1C1 meets PQ at K . assume I is the center of the incircle .
prove IK is parallel to AL
Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.
(Here we denote $XY$ the length of the line segment $XY$.)
Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.
$A$ graph $G$ arises from $G_{1}$ and $G_{2}$ by pasting them along $S$ if $G$ has induced subgraphs $G_{1}$, $G_{2}$ with $G=G_{1}\cup G_{2}$ and $S$ is such that $S=G_{1}\cap G_{2}.$ A is graph is called [i]chordal[/i] if it can be constructed recursively by pasting along complete subgraphs, starting from complete subgraphs. For a graph $G(V,E)$ define its Hilbert polynomial $H_{G}(x)$ to be
$H_{G}(x)=1+Vx+Ex^2+c(K_{3})x^3+c(K_{4})x^4+\ldots+c(K_{w(G)})x^{w(G)},$
where $c(K_{i})$ is the number of $i$-cliques in $G$ and $w(G)$ is the clique number of $G$. Prove that $H_{G}(-1)=0$ if and only if $G$ is chordal or a tree.
Let $ABC$ be a isosceles triangle, with $AB=AC$. A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.
In an acute scalene triangle $ABC$ the bisector of the acute angle between the altitudes $AA_1$ and $CC_1$ meets the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The bisector of the angle $B$ intersects the segment joining the orthocenter of $ABC$ and the midpoint of $AC$ at point $R$. Prove that $P$, $B$, $Q$, $R$ lie on a circle.
Let $ ABC$ be an acute-angled triangle. We consider the equilateral triangle $ A'UV$, where $ A' \in (BC)$, $ U\in (AC)$ and $ V\in(AB)$ such that $ UV \parallel BC$. We define the points $ B',C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.
Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$,
$ r$ being inradius.
Let $ABCDEF$ be a convex hexagon with $\angle A= \angle D$ and $\angle B=\angle E$ . Let $K$ and $L$
be the midpoints of the sides $AB$ and $DE$ respectively. Prove that the sum of the areas of triangles $FAK$, $KCB$ and $CFL$ is equal to half of the area of the hexagon if and only if
\[\frac{BC}{CD}=\frac{EF}{FA}.\]
On the plane is give a broken line $ABCD$ in which $AB = BC = CD = 1$, and $AD$ is not equal to $1$. The positions of $B$ and $C$ are fixed but $A$ and $D$ change their positions in turn according to the following rule (preserving the distance rules given): the point $A$ is reflected with respect to the line $BD$, then $D$ is reflected with respect to the line $AC$ (in which $A$ occupies its new position), then $A$ is reflected with respect to the line $BD$ ($D$ occupying its new position), $D$ is reflected with respect to the line $AC$, and so on. Prove that after several steps $A$ and $D$ coincide with their initial positions.
(M Kontzewich)
Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$.
[i]Ray Li.[/i]